# Direct experimental test of commutation relation via imaginary weak value

June 25, 2021 1:46 pmIn our latest work ^{1}, we tested canonical commutation in an interferometric experiment. The canonical commutation relation is a tenet of quantum theory and Heisenberg’s uncertainty relation is a direct consequence of it. Without loss of generality, consider two non-commuting qubit observables, say and satisfying for all where is a nonzero quanity. Since the product may *not be Hermitian* in general, the traditional von Neumann quantum measurement scheme cannot be carried but the weak measurement sufficed for our purpose. Non-Hermitian observables may be measured using weak measurements. But our scheme here is direct and fundamentally different, as follows. If and are positive-eigenvalue projectors of and respectively, then by using the definition of projection operators and , one has , where is the weak value of the projector for a post-selected state and pre-selected state and is the probability of successful post-selection. If and , the commutation relation reads . By writing , and where , and are the projectors with positive eigenvalue corresponding to the observables , and , respectively. It follows , where is the weak value of given the pre- and post-selected states and , respectively. Thus, the imaginary part of a single weak value is suited to test the commutation relation.

To determine the weak value of the path projector onto path 1, denoted as , an established neutron interferometric setup suited to carry out *weak measurements* (see here for theoretical backgroud), depicted above, is applied. The system is prepared (pre-selected) in the initial path state and post-selected in final state . In order to weakly couple the path degree of freedom to the ancilla, which is represented by the coupled spin-energy degree of freedom, a weak spin rotation via an RF coil is induced in path 1. This RF spin manipulation (see here for theory )leads to an energy-shift by the amount in the spin-flipped parts and can be written as a unitary operator:

where , and are the spin-rotation angle, the frequency of the electromagnetic RF field and an arbitrary phase of this RF field, respectively. The energy-shifted contributions appear due to the off-diagonal terms. The parameter is related to magnetic field strength and for a small value of becomes:

. This results in a time dependent interference pattern of intensity . The weak value for the path projector is a complex number and can therefore be written in the polar form , which is compared to the time dependent interference pattern, expressed in amplitude and phase as , which is depicted below.

By switching off the RF field, two post-selected probabilities and are measured. Finally, the results of all three measurements, that is , and , are combined, for the direct test of the commutation relation, which can be seen below.

1. R. Wagner, W. Kersten, A. Danner, H. Lemmel, A. K. Pan, and S. Sponar *Phys. Rev. Research* **3**, 023243 (2021). ↩